91Ó°ÊÓ

(i) A group \(G\) is centerless if \(Z(G)=1\). Prove that \(S_{n}\) is centerless if \(n \geq 3\). (ii) Prove that \(A_{4}\) is centerless.

(i) A finite group \(G\) with exactly two conjugacy classes has order \(2 .\) (ii) Let \(G\) be a group containing an element of finite order \(n>1\) and exactly two conjugacy classes. Prove that \(|G|=2\). (Hint. There is a prime \(p\) with \(a^{p}=1\) for all \(a \in G\). If \(p\) is odd and \(a \in G\), then \(a^{2}=x a x^{-1}\) for some \(x\), and so \(a^{2 k}=x^{k} a x^{-k}\) for all \(k \geq 1 .\) ) (There are examples of infinite groups \(G\) with no elements of finite order which do have exactly two conjugacy classes.)

(i) Prove, for every \(a, x \in G\), that \(C_{G}\left(a x a^{-1}\right)=a C_{G}(x) a^{-1}\). (ii) Prove that if \(H \leq G\) and \(h \in H\), then \(C_{H}(h)=C_{G}(h) \cap H\).

(i) Let \(k\) be a field. If \(G=G L(n, k)\) and \(T\) is the subgroup of \(G\) of all diagonal matrices, then \(N_{G}(T)\) consists of all the monomial matrices over \(k\). (ii) Prove that \(N_{G}(T) / T \cong S_{n}\) -

\(A_{4}\) is the only subgroup of \(S_{4}\) having order \(12 .\)

Show that the number of conjugacy classes in \(S_{n}\) is the number of partitions of \(n .\)

If \(G \leq S_{n}\) contains an odd permutation, then \(|G|\) is even and exactly half the elements of \(G\) are odd permutations.

(i) Let \(G\) be a group of order \(2^{m} k\), where \(k\) is odd. Prove that if \(G\) contains an element of order \(2^{m}\), then the set of all elements of odd order in \(G\) is a (normal) subgroup of G. (Hint. Consider \(G\) as permutations via Cayley's theorem, and show that it contains an odd permutation.) (ii) Show that a finite simple group of even order must have order divisible by \(4 .\)

Let \(G\) be a finite group containing a subgroup \(H\) of index \(p\), where \(p\) is the smallest prime divisor of \(|G| .\) Prove that \(H\) is a normal subgroup of \(G\).

Let \(G\) be an infinite simple group. (i) Every \(x \in G\) with \(x \neq 1\) has infinitely many conjugates. (ii) Every proper subgroup \(H \neq 1\) has infinitely many conjugates.